Algebra Examples

$4 x^{2} + 12 x - 15 = 5$

Step 1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1

Add $15$ to both sides of the equation.

$4 x^{2} + 12 x = 5 + 15$

Step 1.2

Add $5$ and $15$ .

$4 x^{2} + 12 x = 20$

Step 2

Divide each term in $4 x^{2} + 12 x = 20$ by $4$ and simplify.

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Step 2.1

Divide each term in $4 x^{2} + 12 x = 20$ by $4$ .

$\frac{4 x^{2}}{4} + \frac{12 x}{4} = \frac{20}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $4$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} + \frac{12 x}{4} = \frac{20}{4}$

Step 2.2.1.1.2

Divide $x^{2}$ by $1$ .

$x^{2} + \frac{12 x}{4} = \frac{20}{4}$

Step 2.2.1.2

Cancel the common factor of $12$ and $4$ .

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Step 2.2.1.2.1

Factor $4$ out of $12 x$ .

$x^{2} + \frac{4 (3 x)}{4} = \frac{20}{4}$

Step 2.2.1.2.2

Cancel the common factors.

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Step 2.2.1.2.2.1

Factor $4$ out of $4$ .

$x^{2} + \frac{4 (3 x)}{4 (1)} = \frac{20}{4}$

Step 2.2.1.2.2.2

Cancel the common factor.

$x^{2} + \frac{4 (3 x)}{4 \cdot 1} = \frac{20}{4}$

Step 2.2.1.2.2.3

Rewrite the expression.

$x^{2} + \frac{3 x}{1} = \frac{20}{4}$

Step 2.2.1.2.2.4

Divide $3 x$ by $1$ .

$x^{2} + 3 x = \frac{20}{4}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $20$ by $4$ .

$x^{2} + 3 x = 5$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(\frac{3}{2})}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} + 3 x + {(\frac{3}{2})}^{2} = 5 + {(\frac{3}{2})}^{2}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Apply the product rule to $\frac{3}{2}$ .

$x^{2} + 3 x + \frac{3^{2}}{2^{2}} = 5 + {(\frac{3}{2})}^{2}$

Step 5.1.1.2

Raise $3$ to the power of $2$ .

$x^{2} + 3 x + \frac{9}{2^{2}} = 5 + {(\frac{3}{2})}^{2}$

Step 5.1.1.3

Raise $2$ to the power of $2$ .

$x^{2} + 3 x + \frac{9}{4} = 5 + {(\frac{3}{2})}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $5 + {(\frac{3}{2})}^{2}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Apply the product rule to $\frac{3}{2}$ .

$x^{2} + 3 x + \frac{9}{4} = 5 + \frac{3^{2}}{2^{2}}$

Step 5.2.1.1.2

Raise $3$ to the power of $2$ .

$x^{2} + 3 x + \frac{9}{4} = 5 + \frac{9}{2^{2}}$

Step 5.2.1.1.3

Raise $2$ to the power of $2$ .

$x^{2} + 3 x + \frac{9}{4} = 5 + \frac{9}{4}$

Step 5.2.1.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} + 3 x + \frac{9}{4} = 5 \cdot \frac{4}{4} + \frac{9}{4}$

Step 5.2.1.3

Combine $5$ and $\frac{4}{4}$ .

$x^{2} + 3 x + \frac{9}{4} = \frac{5 \cdot 4}{4} + \frac{9}{4}$

Step 5.2.1.4

Combine the numerators over the common denominator.

$x^{2} + 3 x + \frac{9}{4} = \frac{5 \cdot 4 + 9}{4}$

Step 5.2.1.5

Simplify the numerator.

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Step 5.2.1.5.1

Multiply $5$ by $4$ .

$x^{2} + 3 x + \frac{9}{4} = \frac{20 + 9}{4}$

Step 5.2.1.5.2

Add $20$ and $9$ .

$x^{2} + 3 x + \frac{9}{4} = \frac{29}{4}$

Step 6

Factor the perfect trinomial square into ${(x + \frac{3}{2})}^{2}$ .

${(x + \frac{3}{2})}^{2} = \frac{29}{4}$

Step 7

Solve the equation for $x$ .

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Step 7.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x + \frac{3}{2} = \pm \sqrt{\frac{29}{4}}$

Step 7.2

Simplify $\pm \sqrt{\frac{29}{4}}$ .

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Step 7.2.1

Rewrite $\sqrt{\frac{29}{4}}$ as $\frac{\sqrt{29}}{\sqrt{4}}$ .

$x + \frac{3}{2} = \pm \frac{\sqrt{29}}{\sqrt{4}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Rewrite $4$ as $2^{2}$ .

$x + \frac{3}{2} = \pm \frac{\sqrt{29}}{\sqrt{2^{2}}}$

Step 7.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x + \frac{3}{2} = \pm \frac{\sqrt{29}}{2}$

Step 7.3

Subtract $\frac{3}{2}$ from both sides of the equation.

$x = \pm \frac{\sqrt{29}}{2} - \frac{3}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{29}}{2} - \frac{3}{2}$

Decimal Form:

$x = 1.19258240 \dots, - 4.19258240 \dots$